91Ó°ÊÓ

(a) Find three different Pythagorean triples, not necessarily primitive, of the form 16 , \(y, z\) (b) Obtain all primitive Pythagorean triples \(x, y, z\) in which \(x=40 ;\) do the same for \(x=60\)

(a) Establish that there exist infinitely many primitive Pythagorean triples \(x, y, z\) in which \(x\) and \(y\) are consecutive positive integers. Exhibit five of these. [Hint: If \(x, x+1, z\) forms a Pythagorean triple, then so does the triple \(3 x+2 z+1\), \(3 x+2 z+2,4 x+3 z+2 .]\) (b) Show that there exist infinitely many Pythagorean triples \(x, y, z\) in which \(x\) and \(y\) are consecutive triangular numbers. Exhibit three of these. [Hint: If \(x, x+1, z\) forms a Pythagorean triple, then so does \(\left.t_{2 x}, t_{2 x+1},(2 x+1) z .\right]\)